CTMRG with iterative (Lanczos) solver for SVD

[Yue-Zhengyuan]

Let $\chi$ be the dimension of the CTMRG environment. In the projector finding step of CTMRG, we know that only $\chi$ singular values (and their corresponding left/right singular vectors) are needed. Therefore, instead of doing a full SVD and then truncate, it can greatly improve the speed by using an iterative (Lanczos) algorithm for the SVD step, as noted in the paper (arXiv:2308.12358, particularly when $\chi$ is large; see Fig. 16 there)

Naumann, Jan, et al. "An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation." SciPost Physics Lecture Notes (2024): 086.

It is stated that the algorithm is already in KrylovKit.jl. Can someone demonstrate how to use such an algorithm with CTMRG? I think it just amounts to setting svd_alg = IterSVD() in CTMRG, but it seems that ProjectorAlg has not been implemented for IterSVD. Thanks!

[pbrehmer]

Hey Yue, totally missed that question, sorry! I hope you figured this out in the meantime but in case this is still unclear, here is an example:

using KrylovKit
using TensorKit
using PEPSKit

Dbond = 2
χenv = 16
svd_alg = SVDAdjoint(; fwd_alg=IterSVD(; alg=GKL(; tol=1e-10)))
ctmrg_alg = SimultaneousCTMRG(; svd_alg)
ψ = InfinitePEPS(2, Dbond)
env = leading_boundary(CTMRGEnv(ψ, ℂ^χenv), ψ, ctmrg_alg);

I'll put an example in the docs once we do the documentation overhaul.

[Yue-Zhengyuan]

@pbrehmer Thanks! However, with symmetry or for fermions, I suppose it requires us to specify in advance the dimensions of each sector, and it cannot dynamically change them (similar to FixedSpaceTruncation for usual SVD)?

[pbrehmer]

Yes indeed. Not all truncation schemes are implemented or would even make sense straightaway in the context of IterSVD. Currently, only NoTruncation and TruncationSpace are implemented (so using FixedSpaceTruncation works just fine).

[pbrehmer]

If you're interested, you can find an example usage in test/utility/svd_wrapper.jl!